Question

Find the Remainder when x^100 is divided by x²-3x+2​

Answer 1

Step-by-step explanation:

Given

p(X)=x^100

Divisor f(x)=x²-3x+2

=x²-2x-x+2

=x(x-2)-1(x-2)

=(x-1)(x-2)

Let Quotient be q(x) and Remainder be r

Since Degree of Divisor is 2 so Remainder will be=ax+B

According to Divison Algorithm

p(x)=f(x)×q(x)+r

x^100=(x-1)(x-2)q(x)+ax+b

Putting Value of x=1

1^100=(1-1)(1-2)q(1)+a×1+b

1=0×-1×q(1)+a+b

a+b=1. (Equation 1)

Putting Value of x=2

2^100=(2-1)(2-2)q(1)+a×2+b

2^100=2a+b. (Equation 2)

Subtracting Equation 2 from 1

2a+b-(a-b)=2^100-1

a=2^100-1

a+b=1 (Equation 1)

b=1-a

b=1-(2^100-1)

b=2-2^100

Remainder=ax+B

(2^100-1)x+(2-2^100)